Question

1/4(cot^4 30°- cosec^4 60°)+3/2(sec^2 45°-tan^2 30°)-5cos^2 60°​

Answer 1

We have to find the value of

1/4 (cot⁴30° - cosec⁴60°) + 3/2 (sec²45° - tan²30°) - 5cos²60°

we know, cot30° = √3

cosec60° = 2/√3

sec45° = √2

tan30° = 1/√3

cos60° = 1/2

now, 1/4 (cot⁴30° - cosec⁴60°) + 3/2 (sec²45° - tan²30°) - 5cos²60°

= 1/4 [(√3)⁴ - (2/√3)⁴] + 3/2 [(√2)² - (1/√3)²] - 5(1/2)²

= 1/4 [9 - 16/9 ] + 3/2 [2 - 1/3 ] - 5/4

= 1/4 × (81 - 16)/9 + 3/2 (6 - 1)/3 - 5/4

= 1/4 × 65/9 + 3/2 × 5/3 - 5/4

= 65/36 + 5/2 - 5/4

= (65 + 90 - 45)/36

= (155 - 45)/36

= 110/36

= 55/18

Therefore the value of 1/4 (cot⁴30° - cosec⁴60°) + 3/2 (sec²45° - tan²30°) - 5cos²60° is 55/18.